288 research outputs found
A Model for Optimal Human Navigation with Stochastic Effects
We present a method for optimal path planning of human walking paths in
mountainous terrain, using a control theoretic formulation and a
Hamilton-Jacobi-Bellman equation. Previous models for human navigation were
entirely deterministic, assuming perfect knowledge of the ambient elevation
data and human walking velocity as a function of local slope of the terrain.
Our model includes a stochastic component which can account for uncertainty in
the problem, and thus includes a Hamilton-Jacobi-Bellman equation with
viscosity. We discuss the model in the presence and absence of stochastic
effects, and suggest numerical methods for simulating the model. We discuss two
different notions of an optimal path when there is uncertainty in the problem.
Finally, we compare the optimal paths suggested by the model at different
levels of uncertainty, and observe that as the size of the uncertainty tends to
zero (and thus the viscosity in the equation tends to zero), the optimal path
tends toward the deterministic optimal path
Space-time FLAVORS: finite difference, multisymlectic, and pseudospectral integrators for multiscale PDEs
We present a new class of integrators for stiff PDEs. These integrators are
generalizations of FLow AVeraging integratORS (FLAVORS) for stiff ODEs and SDEs
introduced in [Tao, Owhadi and Marsden 2010] with the following properties: (i)
Multiscale: they are based on flow averaging and have a computational cost
determined by mesoscopic steps in space and time instead of microscopic steps
in space and time; (ii) Versatile: the method is based on averaging the flows
of the given PDEs (which may have hidden slow and fast processes). This
bypasses the need for identifying explicitly (or numerically) the slow
variables or reduced effective PDEs; (iii) Nonintrusive: A pre-existing
numerical scheme resolving the microscopic time scale can be used as a black
box and easily turned into one of the integrators in this paper by turning the
large coefficients on over a microscopic timescale and off during a mesoscopic
timescale; (iv) Convergent over two scales: strongly over slow processes and in
the sense of measures over fast ones; (v) Structure-preserving: for stiff
Hamiltonian PDEs (possibly on manifolds), they can be made to be
multi-symplectic, symmetry-preserving (symmetries are group actions that leave
the system invariant) in all variables and variational
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